Pro babble or pro bob level

Looking a Table 1, there are eleven ways to roll a specific number, such as 1. 

Question:

Are there, twice the number or twenty-two ways to roll

                                                                          one of any two different numbers? 

No.

There are only twenty ways, since 1-2 and 2-1 cannot be counted twice.

Similarly, there are twenty-seven ways to roll one of three specified numbers,

 thirty-two ways to roll one of four, and thirty-five ways to roll one of five.

Table 2
Probability of Entering from the Bar

Number of points open	Ways to come in	Chance of coming in	Odds in favor or against
5	35	97%	35 to 1 in favor
4	32	89%	8 to 1 in favor
3	27	75%	3 to 1 in favor
2	20	56%	5 to 4 in favor
1	11	31%	25 to 11 against

Table 2 merely reflect the preceding, since the “number of points open” corresponds to the number of different numbers you can roll. Thus if you have a man on the bar and there are five points open in your opponent’s inner board, you want to know the probability of rolling any of five specified numbers.

The ways the can be named

as the possible ways 

that can be 

named

Be Gin in G

with the idea

of E

as Three 

one more than two

and one less than two pair

then one less than a few

followed by five 

subsequently by six

silently by seven

then down 

the line

back

to

1

1 and 1 are the only 

in t g ers

that fit into two

Two is the lowest sum of a pair of dice

of which the possibility of 

x is y

 1  0

2  1

 3  2 

4  3

5  4

6  5

7  6

8  5

9  4

10  3

11  2

12  1

AKA

Sum

Possible ways

1  2  3  4  5  6  7  8  9  10  11  12

0  1  2  3  4  5  6  5  4   3   2    1

7 =       6

6 - 8    5

5 - 9    4

4  - 10   3

3 - 11   2

2  - 12   1

1   =     0

The ways of seven are six

2+5

3+4

4+3

5+2

6+1

1+6

The ways of six are five

2+4

3+3

4+2

5+1

1+5

The ways of eight are five

2+6

3+5

4+4

5+3

6+2

The ways of five are four

2+3

3+2

4+1

1+4

The ways of nine are four

3+6

4+5

5+4

6+3

The ways of four are three

1+3

2+2

3+1

The ways of ten are three

4+6

5+5

6+4

The ways of eleven are two

5+6

6+5

The way of twelve is one

6+6

Rolling At Least One of a Number with two balanced Dice

There are a total of six outcomes, corresponding to each of the integers from 1 to 6. Thus each number has a probability of 1/6 of occurring.

(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (2, 1), (2, 3), (2, 4), (2, 5), (2, 6)

(2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (2, 1), (2, 3), (2, 4), (2, 5), (2, 6), (1, 2)

When we roll two dice, each die is independent of the other. If we keep track of the order of what number occurs on each of the dice, then there are a total of 6 x 6 = 36 equally likely outcomes. Thus 36 is the denominator for all of our probabilities and any particular outcome of two dice has a probability of 1/36.

The probability of rolling two dice and getting at least one of a number from 1 to 6 is straightforward to calculate.

 If we wish to determine the probability of rolling at least one 2 with two dice, 

we need to know how many of the 36 possible outcomes include at least one 2. The ways of doing this are:

(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2), (2, 1), (2, 3), (2, 4), (2, 5), (2, 6)

(1, 3), (2, 3)...

(1, 4), (2, 4), (3, 4)...

(1, 5), (2, 5), (3, 5)...

(1, 6), (2, 6), (3, 6)...

Thus there are 11 ways to roll at least one 2 with two dice, and the probability of rolling at least one 2 with two dice is 11/36.

There is nothing special about 2 in the preceding discussion. For any given number n from 1 to 6:

• There are five ways to roll exactly one of that number on the first die.
• There are five ways to roll exactly one of that number on the second die.
• There is one way to roll that number on both dice.

Therefore there are 11 ways to roll at least one n from 1 to 6 using two dice. The probability of this occurring is 11/36.